#6441376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,33)(21,27)(22,38)(23,29)(24,28)(26,30)(31,44)(32,43)(34,47)(35,46)(36,48)(37,41)(39,42)(40,45);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,23,32)(18,25,29,43)(20,34,30,40)(21,35,37,36)(22,31,24,39)(26,45,33,47)(27,46,41,48)(28,42,38,44);(10,14)(13,16)(17,20,23,30)(18,26,29,33)(19,21,32,37)(22,36,24,35)(25,41,43,27)(28,48,38,46)(31,40,39,34)(42,45,44,47);(1,4,5,3)(2,6,7,8)(17,21)(18,27)(19,30)(20,32)(22,34)(23,37)(24,40)(25,26)(28,45)(29,41)(31,36)(33,43)(35,39)(38,47)(42,46)(44,48);(17,22,23,24)(18,28,29,38)(19,31,32,39)(20,35,30,36)(21,34,37,40)(25,42,43,44)(26,46,33,48)(27,45,41,47)

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348572);